Let `y=ax^2+bx+c` be equation that defines our parabola.

Coordinates of vertex are:` (-b/(2a),-D/(4a))`

Coordinates of focus are:`(-b/(2a),(1-D)/4a)`

Where `D` is discriminant `D=b^2-4ac`. Note that `x` coordinates are the same, which is to be expected because both vertex anf focus are points on the axis of symmetry.

Therefore...

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Let `y=ax^2+bx+c` be equation that defines our parabola.

Coordinates of vertex are:` (-b/(2a),-D/(4a))`

Coordinates of focus are:`(-b/(2a),(1-D)/4a)`

Where `D` is discriminant `D=b^2-4ac`. Note that `x` coordinates are the same, which is to be expected because both vertex anf focus are points on the axis of symmetry.

Therefore focal length (distance between focus and vertex) is `f=|(1-D)/(4a)-(-D)/(4a)|=1/(4|a|)`.

So if `f` is very large then `|a|` must be very small because from above formula we have` ` `|a|=1/(4f)`

So as `f` gets bigger, `|a|` becomes smaller so we have `y=ax^2+bx+c` where `a` is very small. If `a` were to become infinitely small, our equation would become `y=bx+c` which is equation of a line.

So, what happens with parabola as focal length becomes very great?

**1.** Two branches of parabola become more and more distant (i.e. parabola is becoming wider and wider)

**2.** Ultimately as focal length becomes infinitaly great parabola tends to look like line parallel to directrix.

Graph shows parabolas for different `a` (so different `f` ).

From purple, blue, green, yellow, orange, red and black `f` is equal to `1/40,1/20,1,5/4,5/2,25`

For more on parabola see the link below.